Conformal symplectic and constraint-preserving model order reduction of constrained conformal Hamiltonian systems

Abstract

Dissipative and constrained dynamical systems model various physical phenomena comprising damping and constraining forces. These forces give rise to specific geometric properties of the dynamical system's governing equations. In this paper, we reduce and solve high-dimensional parameterized constrained and damped nonlinear differential equations with structure-preserving algorithms. These algorithms preserve the constrained equation's geometric properties, such as conformal symplecticness and phase-space, through model order reduction and numerical integration. We develop a reduced model based on an orthosymplectic reduced basis and establish the well-posedness and conformal symplecticness of the reduced model. Moreover, we derive a priori error bounds for state space and constraints errors. Subsequently, we utilize two RATTLE-based integrators to simulate the full and reduced models while preserving their geometric properties. Numerical experiments on constrained damped oscillators and a constrained lattice grid justify theoretical results and demonstrate the advantages of the structure-preserving computational methods for constrained dynamical systems.

Keywords

• Constrained Hamiltonian systems
• dissipation
• symplectic integration
• symplectic reduced basis
• error estimation
• 65P10